Quantifying drug interactions in drug combination studies and classifying the interactions into categories of synergy, additivity, or antagonism are of interest to many researchers. We appreciate the comments from Dr. Chou and the opportunity to address this important issue. For illustration, we use the simplest setting: the combination of two drugs. The Loewe additivity model has been widely used as a reference model when the combined effect of two drugs is additive. The model can be written as in Eq. 1:

where d1 and d2 are the respective combination doses of drug 1 and drug 2 that yield an effect y, and Dy,1 and Dy,2 are the corresponding single doses for drug 1 and drug 2 that result in the same effect. When Eq. 1 holds, we conclude that the combined effect of the two drugs is additive. One can easily check that Eq. 1 holds in a “sham” experiment when the two drugs are identical. The Loewe additivity model is rooted in the use of the isobologram approach, which was first proposed by Fraser in the 1870s as a graphical tool for assessing drug interaction (1, 2). Loewe and Muischnek provided explicit mathematical and geometric derivations for this model in 1926 (3). The explicit writing for the formula was given on page 179 in a subsequent article by Loewe (4). A more detailed exposition and expansion of this approach can be found in Loewe's 1953 landmark article titled “The problem of synergism and antagonism of combined drugs” (5). Since then, many authors have attributed Eq. 1 to Loewe (6–14).

Based on Eq. 1, the interaction index, defined in Eq. 2, can be used to classify drug interaction as synergistic, additive, or antagonistic.

The interaction index has been discussed by several authors, such as Berenbaum (9), Tallarida (14), and Meadows and colleagues (15). The definition of the interaction index coincides with Chou and Talalay's definition of the combination index for mutually exclusive drugs, published in 1984 (16). The field of drug combination research spans more than 100 years and has been addressed within many disciplines. Chou and Talalay's seminal article has made important contributions and has been widely cited in the literature. However, this article was not the first nor the only one that generates concepts supporting Eqs. 1 and 2 and applies these equations to study drug interactions. Lee and colleagues directed the reader to reviews of these methods, with proper citation of the appropriate references therein (17), and derived the confidence interval estimation for the interaction index and compared its performance with four methods based on response surface models. Although several equations are indeed similar to equations that appeared in articles from Dr. Chou, they also appear in numerous other publications by other researchers addressing this subject.

In their data analysis, Jia and colleagues used “proportion surviving” to measure drug activity (18). Dr. Chou's 1984 work uses “fraction affected” or “fraction killed” to measure drug activity. Clearly, these two measures are complementary to each other. In Eqs. 1 or 2, the combination dose (d1, d2) and the single agent doses Dy,1 and Dy,2 all correspond to “an effect y.” It does not matter that the effect “y” is defined as the fraction killed or fraction survived. In other words, the doses resulting in 30% cell death are the same as those resulting in 70% cell survival. Thus, the doses resulting in kill fraction K are the same as those resulting in survived fraction 1-K, and the resulting interaction index is the same taking either approach. Expandng this basic concept, the results of assessing the mode of drug interaction do not and should not depend on whether the drug effect is measured by fraction killed or fraction survived. A more detailed algebraic derivation is provided online in the Supplementary Material, confirming that the two approaches are not in conflict. In fact, identical results, such as the synergistic effect of mithramycin and tolfenamic acid, are found to inhibit pancreatic cell proliferation by using either method (16, 17).

Dr. Chou stated reservations regarding the use of a probabilistic/statistical approach to assess drug interaction, asking, “Is drug combination synergy quantification a mass-action law issue or a statistical issue?” He has asserted in an earlier publication that “…synergism is basically a physiochemical mass-action law issue, not a statistical issue…” and that one must “…[d]etermine synergism with CI [combination index] values, not with P values” (19). We agree that the reference model for assessing drug interaction can be obtained by mathematics based on an understanding of the drugs' mechanisms of action. When the drug effect is measured without error (i.e., both the drug dose and its activity as measured by the proportion of cells that were killed or which survived are 100% accurate), no statistics are required to assess the combination drug effect. However, stochastic variations inherent in biological heterogeneity and measurement error generate two major sources of uncertainty. In examples where uncertainty exists, statistics must play a critical role. In our work, we have constructed confidence intervals under the various models used to assess drug interaction (17, 20–22). If the upper end of the confidence interval of the interaction index is smaller than 1, synergy is claimed. Conversely, if the lower end of the confidence interval of the interaction index is greater than 1, antagonism is reached. Otherwise, one will conclude that the drug interaction is additive. In summary, although the effect of a drug combination may follow a mass-action law, its assessment requires statistical approaches to properly manage existing uncertainty.